以下のコードを試してみてください。目的の変更が行われています
<div id="material">
<input type="radio" name="materials" value="glass" style="vertical-align: middle" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'glass') {echo "checked";}} ?>/>
<label for="material">Glass</label> <br/>
<input type="radio" checked="checked" name="materials" style="vertical-align: middle" value="plastic" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'plastic') {echo "checked";}} ?>/>
<label for="material" >Plastic (Organic)</label>
</div>
更新しました
<form method="post" >
<div id="material">
<?php if(isset($_POST['materials'])) {
if($_POST['materials'] == 'glass'){
?>
<input type="radio" checked="checked" name="materials" value="glass" style="vertical-align: middle" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'glass') {echo "checked";}} ?>/>
<label for="material">Glass</label> <br/>
<input type="radio" name="materials" style="vertical-align: middle" value="plastic" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'plastic') {echo "checked";}} ?>/>
<label for="material" >Plastic (Organic)</label>
<?php }
else { ?>
<input type="radio" name="materials" value="glass" style="vertical-align: middle" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'glass') {echo "checked";}} ?>/>
<label for="material">Glass</label> <br/>
<input type="radio" checked="checked" name="materials" style="vertical-align: middle" value="plastic" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'plastic') {echo "checked";}} ?>/>
<label for="material" >Plastic (Organic)</label>
<?php }
} else { ?>
<input type="radio" name="materials" value="glass" style="vertical-align: middle" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'glass') {echo "checked";}} ?>/>
<label for="material">Glass</label> <br/>
<input type="radio" checked="checked" name="materials" style="vertical-align: middle" value="plastic" onchange="this.form.submit()" <?php if(isset($_POST['materials'])){ if($_POST['materials'] == 'plastic') {echo "checked";}} ?>/>
<label for="material" >Plastic (Organic)</label>
<?php } ?>
</div>
<input type="submit" name="submit">
</form>